Isn't this just an overreaction, a correspondent asks? In this modern age, why can't they just operate the trading floor remotely?
The answer is that the NYSE considered it, and rejected the idea. The members weren't comfortable with the possibility that there would be glitches in the electronic system (as has happened before). "Glitches", in the context of international finance, can mean "millions of dollars lost and a huge pain in the ass for everyone as trades have to be unwound".
And given the problems with the subway system, there was no way that they could operate the physical floor. The subway systems are one of New York's great achievements--they not only provide a greener method of travel than individual cars, but also allow you to move far more workers into a small downtown area than would be possible by automobile. The downside is that when the subways aren't operating, there's no real substitute. Even if the stock exchange had opened, financial firms would have had a hell of a time getting all their workers in place to trade.
Financial firms do have disaster recovery sites located a safe distance from their main offices, where business can be done in at least a skeleton fashion if headquarters is shut down. But those sites have to be located within a reasonable distance of the main office so that your workers can get there. When you're dealing with a storm that stretches from DC to Boston, no conceivably located disaster recovery site will do you any good.
Moreover, while the internet has certainly shrunk distances, it hasn't yet made them irrelevant. Having spent a fair number of years doing IT work on trading floors, it's hard for me to picture email and IM as a good substitute. For the same reason that meetings are more efficient than endless email threads, not all business can be conducted by telecommute.
Maybe someday we'll have a financial system like some sort of neural network, that can make geography not matter. But even that wouldn't have done much good yesterday. Half the workers who would have been trading on that advanced network, or processing orders, or what have you, have probably lost power and/or internet connectivity. We have gone a long way towards mitigating the ravages of nature. But it's still a whole lot more powerful than we are.